Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2401.09222

Local Trajectory Variation Exponent (LTVE) for Visualizing Dynamical Systems

Yun Chen Tsai·Shingyu Leung
View original source ↗Read source document ↗Export metadata ↓

Abstract

The identification and visualization of Lagrangian structures in flows plays a crucial role in the study of dynamic systems and fluid dynamics. The Finite Time Lyapunov Exponent (FTLE) has been widely used for this purpose; however, it only approximates the flow by considering the positions of particles at the initial and final times, ignoring the actual trajectory of the particle. To overcome this limitation, we propose a novel quantity that extends and generalizes the FTLE by incorporating trajectory metrics as a measure of similarity between trajectories. Our proposed method utilizes trajectory metrics to quantify the distance between trajectories, providing a more robust and accurate measure of the LCS. By incorporating trajectory metrics, we can capture the actual path of the particle and account for its behavior over time, resulting in a more comprehensive analysis of the flow. Our approach extends the traditional FTLE approach to include trajectory metrics as a means of capturing the complexity of the flow.

Explore related subjects

math.DSphysics.flu-dyn

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yun Chen Tsai, Shingyu Leung. 2024-01-17. Local Trajectory Variation Exponent (LTVE) for Visualizing Dynamical Systems. https://arxiv.org/abs/2401.09222

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗